The formation of single-crystal diamond membranes is an important prerequisite for the fabrication of high-quality optical cavities in this material. Diamond membranes fabricated using lift-off processes involving the creation of a damaged layer through ion implantation often suffer from residual ion damage, which severely limits their usefulness for photonic structures. The current work demonstrates that strategic etch removal of the most highly defective material yields thin, single-crystal diamond membranes with strong photoluminescence and a Raman signature approaching that of single-crystal bulk diamond. These optically-active membranes can form the starting point for fabrication of high-quality optical resonators. ; Comment: To appear in APL